International A and AS Level Mathematics Pure Mathematics 1

[Pages:37]Cambridge

International A and AS Level Mathematics

Pure Mathematics 1

Sophie Goldie Series Editor: Roger Porkess

i7 HODDER EDUCATION AN HACHETTE UK COMPANY



Questions from the Cambridge International Examinations A & AS level Mathematics papers are reproduced by permission of University of Cambridge International Examinations.

Questions from the MEI A & AS level Mathematics papers are reproduced by permission of OCR.

We are grateful to the following companies, institutions and individuals you have given permission to reproduce photographs in this book. page 106, ?Jack Sullivan/ Alamy; page 167, ? RTimages/Fotolia; page 254, ? Hunta/Fotolia; page 258, ? Olga Iermolaieva/ Fotolia

Every effort has been made to trace and acknowledge ownership of copyright. The publishers will be glad to make suitable arrangements with any copyright holders whom it has not been possible to contact.

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Much of the material in this book was published originally as part of the MEI Structured Mathematics series. It has been carefully adapted for the Cambridge International A & AS level Mathematics syllabus.

The original ME! author team for Pure Mathematics comprised Catherine Berry, Bob Francis, Val Hanrahan, Terry Heard, David Martin, Jean Matthews, Bernard Murphy, Roger Porkess and Peter Seeker.

?ME!, 2012

First published in 2012 by Hodder Education, a Hachette UK company, 338 Euston Road London NW! 3BH

Impression number 5 4 3 2 I

Year

2016 2015 2014 2013 2012

All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6-10 Kirby Street, London ECIN 8TS.

Cover photo by ? Joy Fera/Fotolia Illustrations by Pantek Media, Maidstone, Kent Typeset in IO.Spt Minion by Pantek Media, Maidstone, Kent Printed in Dubai

A catalogue record for this title is available from the British Library

ISBN 978 1444 14644 8



Contents

Key to symbols in this book

vi

Introduction

vii

The Cambridge A & AS Level Mathematics 9709 syllabus

viii

Chapter 1

Algebra

1

Background algebra

1

Linear equations

6

Changing the subject of a formula

10

Quadratic equations

12

Solving quadratic equations

17

Equations that cannot be factorised

20

The graphs of quadratic functions

22

The quadratic formula

25

Simultaneous equations

29

Inequalities

34

Chapter 2

Co-ordinate geometry

38

Co-ordinates

38

Plotting, sketching and drawing

39

The gradient of a line

39

The distance between two points

41

The mid-point of a line joining two points

42

The equation of a straight line

46

Finding the equation of a line

49

The intersection of two lines

56

Drawing curves

63

The intersection of a line and a curve

70

Chapter 3

Sequences and series

75

Definitions and notation

76

Arithmetic progressions

77

Geometric progressions

84

Binomial expansions

95

Chapter 4 Chapter 5

Chapter 6 Chapter 7

Functions The language of functions Composite functions Inverse functions

Differentiation The gradient of a curve Finding the gradient of a curve Finding the gradient from first principles Differentiating by using standard results Using differentiation Tangents and normals Maximum and minimum points Increasing and decreasing functions Points of inflection The second derivative Applications The chain rule

Integration Reversing differentiation Finding the area under a curve Area as the limit of a sum

Areas below the x axis

The area between two curves The area between a curve and the y axis The reverse chain rule Improper integrals Finding volumes by integration

Trigonometry Trigonometry background Trigonometrical functions Trigonometrical functions for angles of any size The sine and cosine graphs The tangent graph Solving equations using graphs of trigonometrical functions Circular measure The length of an arc of a circle The area of a sector of a circle Other trigonometrical functions

106 106 112 115

123 123 124 126 131 134 140 146 150 153 154 160 167

173 173 179 182 193 197 202 203 206 208

216 216 217 222 226 228 229 235 239 239 244

Chapter 8

Vectors

254

Vectors in two dimensions

254

Vectors in three dimensions

258

Vector calculations

262

The angle between two vectors

271

Answers

280

Index

310

Key to symbols in this book

0 This symbol means that you want to discuss a point with your teacher. If you are

working on your own there are answers in the back of the book. It is important, however, that you have a go at answering the questions before looking up the answers if you are to understand the mathematics fully.

This symbol invites you to join in a discussion about proof. The answers to these questions are given in the back of the book.

This is a warning sign. It is used where a common mistake, misunderstanding or tricky point is being described.

This is the ICT icon. It indicates where you could use a graphic calculator or a computer. Graphical calculators and computers are not permitted in any of the examinations for the Cambridge International A & AS Level Mathematics 9709 syllabus, however, so these activities are optional.

This symbol and a dotted line down the right-hand side of the page indicates material that you are likely to have met before. You need to be familiar with the material before you move on to develop it further.

e This symbol and a dotted line down the right-hand side of the page indicates

material which is beyond the syllabus for the unit but which is included for completeness.

Introduction

This is the first of a series of books for the University of Cambridge International Examinations syllabus for Cambridge International A & AS Level Mathematics 9709. The eight chapters of this book cover the pure mathematics in AS level. The series also contains a more advanced book for pure mathematics and one each for mechanics and statistics.

These books are based on the highly successful series for the Mathematics in Education and Industry (MEI) syllabus in the UK but they have been redesigned for Cambridge users; where appropriate new material has been written and the exercises contain many past Cambridge examination questions. An overview of the units making up the Cambridge International A & AS Level Mathematics 9709 syllabus is given in the diagram on the next page.

Throughout the series the emphasis is on understanding the mathematics as well as routine calculations. The various exercises provide plenty of scope for practising basic techniques; they also contain many typical examination questions.

An important feature of this series is the electronic support. There is an accompanying disc containing two types of Personal Tutor presentation: examination-style questions, in which the solutions are written out, step by step, with an accompanying verbal explanation, and test yourself questions; these are multiple-choice with explanations of the mistakes that lead to the wrong answers as well as full solutions for the correct ones. In addition, extensive online support is available via the MEI website, .uk.

The books are written on the assumption that students have covered and understood the work in the Cambridge IGCSE syllabus. However, some of the early material is designed to provide an overlap and this is designated 'Background'. There are also places where the books show how the ideas can be taken further or where fundamental underpinning work is explored and such work is marked as 'Extension'.

The original MEI author team would like to thank Sophie Goldie who has carried out the extensive task of presenting their work in a suitable form for Cambridge International students and for her many original contributions. They would also like to thank Cambridge International Examinations for their detailed advice in preparing the books and for permission to use many past examination questions.

Roger Porkess Series Editor

The Cambridge A & AS Level Mathematics syllabus

Cambridge IGCSE

Mathematics

ALevel Mathematic?

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